Rings of Smooth Functions and Their Localizations, I

JOURNAL

OF ALGEBRA

9,

324-336

(1986)

Rings of Smooth Functions and Their Localizations, I I. MOERDIJK

AND

G. E. REYES

Mathematisch Instituut, Univ. van Amsterdam, Roettersstraat 15, Amsterdam 1, 1018 WB, The Netherlands and DPpartement de Mathkmatiques, Universitk de MontrPal, Mont&al, Canada Communicated by Saunders MacLane Received

April

1, 1984

Several types of rings of smooth functions, such as differentiable algebras and formal algebras, occupy a central position in singularity theory and related subjects. In this series of papers we will be concerned with a larger class of rings of smooth functions, which would play a role in Differential Geometry similar to the role played by commutative rings or k-algebras in Algebraic Geometry. This larger class of rings is obtained from rings of smooth functions on manifolds by dividing by ideals and taking filtered colimits. The original motivation to introduce and study P-rings was to construct topos-models for synthetic differential geometry (SDG). The program of SDG (see, e.g., Kock [ 11) was proposed by F. W. Lawvere, and it was in this context that C--rings first appeared explicitly in the literature (see, e.g., Reyes and Wraith [14] and Dubuc [2]). These toposes which provide models for SDG are constructed in a way similar to the toposes occurring in algebraic geometry, but with k-algebras replaced by C”-rings. In particular, the C”-analogue of the Zariski topos, the so-called smooth Zariski topos of Moerdijk and Reyes [ 111 contains a category of “smooth schemes,” just as the usual Zariski topos contains the schemes of algebraic geometry (see Demazure and Gabriel [ 11). Despite this original motivation from SDG, P-rings and their schemes can be studied by themselves, and independently from topos theory in general, and topos-models for SDG in particular. In these two papers, we will start to explore this independent line of development of the theory of P-rings. This can make the connection with algebraic geometry stronger, since the usual presentation of the relation between algebra and geometry takes place at the level of schemes, rather than toposes. The organization of this paper and its sequel, part 11, is as follows. 324 0021-8693/86

$3.00

Copyright 0 1986 by Academic Press, Inc. All rights of reproduclmn in any form reserved.

RINGSOF SMOOTHFUNCTIONS

325

In the first section of this paper, we recall the definition of the category of C”-rings and C~-homomorphisms, we introduce some notation, and collect some basic facts. In Section 2, we study Cm-rings which are local (i.e., have a unique maximal ideal). It will be shown, for example, that any P-domain is a local ring, that every local P-ring is Henselian, and that every P-field is real closed. In part II, written with Ngo Van QuC, these results will be used to define and study the spectrum of a Cm-ring. The two main ingredients are the theorem of Mufioz and Ortega (Theorem 1.3 of this paper), which will enable us to give a coherent axiomatization of the notion of a localization of a P-ring (in an arbitrary Grothendieck topos), and the notion of a P-radical prime ideal (introduced in Section 2 of this paper), which allows us to give an explicit description of the spectrum of a C--ring (in the case of Sets).

1. BASIC PROPERTIESOF Coo-RINGS

As we said above, the notion of a C”-ring stems from the program of synthetic differential geometry. As such, P-rings do not occur explicitly in the classical literature, but the main examples do. Consequently, although the statements of some of the basic facts about Cm-rings seem new, their proofs are either known or easily derivable from known techniques in classical analysis (see e.g., Malgrange [9] and Tougeron ClS]). In this section, we will introduce the notation, and list a few basic facts about C”rings that we will need later on. For more information about C”-rings, the reader is referred to Dubuc [3), Kock [7], Moerdijk & Reyes [ 121. In this paper, ring means commutative ring with unit element. Let C” be the category whose objects are the euclidean spaces W, n&O, and morphisms are all smooth maps. A P-ring is a finite product-preserving functor A: C” -+ Sets. More generally, a O-ring in a topos 6” is a finite product preserving functor C” --P8’. Homomorphisms of P-rings, or C”homomorphisms, are just natural transformations. If A: C” --, Sets (or C” + &) is a P-ring, we will also write A for “the underlying set” A(H). So a C”-ring is a set (or an object of &‘) A in which we can interpret every smooth map Iw” -J W’ as a map A” --Pi A” (in a functorial way), and a P-homomorphism cp:A, + A, is a function cp of the underlying sets which preserves these interpretations, i.e., AZ(f) 0qf’ = V”A, coEvery C”-ring is in particular an H-algebra, and every C”homomo~hism is a morphism of ~-algebras.

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The free P-ring on n generators is the ring Ccu(Rn) of smooth functions KY+ R (the projections xi ,..., x, being the generators), and the C”structure on P(lR’) is defined by composition. By Hadamard’s lemma, any (ring-theoretic) ideal Z in P(lV) is a Cm-congruence, i.e., there is a well-defined P-ring structure on the quotient Cn,(Rn)/Z which makes the projection P( Rn) + Cm( R”)/Z into a P-homomorphism. Filtered colimits of P-rings are constructed as filtered colimits of commutative rings. So if E is any set, the free P-ring with E as a set of generators is P(R”)

:=l&

{C?(R”)ID

al-mite subset of E},

that is, C~(R”) is the ring of functions [WE-+ [w which smoothly depend on finitely many variables only. So any P-ring is isomorphic to one of the form P(W”)/Z. Observe that from this representation it is clear that every P-ring A is formally real (i.e., Vu, ,..., a, E A: 1 + CaT is invertible). Let us recall a lemma of Whitney’s: 1.1.LEMMA. Every closed set Fc R” is the zeroset of a smooth function j UP-+ [0, 11, i-e, F=Z(f)= {xlf(x)=O}. The complement of Z(f) will be denoted by U,. If U c R” is open and is such that U= U,, then f is said to be a characteristic function for U. If E is any set, a subset Fc RE is called a zeroset if there exists and f~ P(R”) such that F= Z(f). Thus Fc [WEis a zeroset iff there exists a finite D c E and a closed PC RD such that F= ~6’ (P), where rrD: [WE-+ RD is the projection. Coproducts of P-rings exist, and the coproduct of A and B in the category of P-rings is denoted by A @ coB. In fact, it suffices to show that coproducts of finitely generated P-rings exist, and here we have the formula

f E P(lV)

cy R”)/Z@ mcy W)/.zE Crn(If-Yx W)/(Z, J), where (Z, J) is the ideal generated by {fi rci lf~ Z} u {g 0rrn21g E .Z}. If A is a P-ring, Art] denotes the ring of polynomials with coefficients in A, i.e., the solution of freely adjoining an element to A in the category of rings. There is also the construction of freely adjoining an element to A in the category of P-rings, which will be denoted by A( t}. So

327

RINGS OF SMOOTH FUNCTIONS

since C”(R) is free on one generator, and if A z Coo(R”)/Z, A(tjrC”(Wx

~)/(Z(x)),

where (Z(x)) is the ideal of functions f(x, t) E P’( [WEx R) generated by the functions g(x) E I. An element p(t) E A { t} can indeed be regarded as a “smooth polynomial” A + A, i.e., p(t) induces a map A -+ A by composition: given a EA, a corresponds to a map C”(R) +a A, and p(a) is defined as the composite Cm(R)

JJ + A@,C”(iR)

“,=‘+A,

or rather as the element of A corresponding to [ 1, a] op. Of course, this is just substituting a for t: if A = C~~~~)/Z, and p is represented by p(x, t) E C”(RE x W), a by a(x) E C”(RE), then p(a) is represented by p(x, a(x)) E CVW. If CIEA, A(a-‘)

denotes the solution of universally inverting a in the category of P-rings. So A (a-” f z A(t)/( t. a - 1). This is not the ring of fractions with some power of a as denominator, but the implicit function theorem yields 1.2.&tOPOSITION.

Zf A = Cm( [WE)/1andff

CW( [WE)represents an element

fe A, then

Here U,-= {x E [WE1f(x) # 0} and Cm( Uf) is the ring of smooth functions on U, depending ora finitely many coordinates, while (Zl Ur) is the ideal of functions generated by the restrictions g / U,, g E 1.

In particular, iffEC”(W), Cm(Rn)(f-l~~CCm(Uf), the ring of smooth functions U,+ R. As said, not every smooth function g: U,-+ 04 is of the form h/f m for some h E P’(W) and some m (i.e., adjoining an inverse for f is not the same for P-rings and for commutative rings). A result that will play a key role, especially in part II (Moerdijk, Qu& Reyes, to appear) is the folfowing theorem, due to Muiioz and Ortega [IO]: 1.3.THEOREM. h,kEC?(R”) Proof

Let UC 08” be open, and g E Cm(U). with U,=Uandg*klU-h\U.

Then there are

(sketch) Let (a, > be a sequenceof (smooth) functions such that

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AND REYES

if x$ U, 0, c: U (each n) and U = U, lien. Define g, by g,,(x)=ctn(x).g(x) g,(x) = 0 if XE U. Now if (p,} is an increasing sequence of seminorms defining the (Frechet-) topology on Cm( KY’), we can put h=1C,.og,-2+Y1 +p,fQ-‘(1 +p,k,)l-’ and k=Cn20G7Y1 + P,wr”u +Pn(&))Y. I For a (Y-ring A (not necessarily finitely generated) this can be rephrased as 1.4(i), while (ii) follows from Lemma 1.1. 1.4,THEOREM. {Algebraic reformulation of 1.1 and 1.3) Let A be uny P-ring, and a~ A. Let q: A -3 A{a-‘> be the universal P-homomorphism, Then (i) VbEAfa-‘) 3c,dEA(b*q(c)=n(d)&q(c)~U(A(a-1jf), for any ring R, U(R) = (r E R 1r is invertible 1; (ii) VbEA(q(b)=O=!kEA (~(c)~U(A~u-‘~)&c.b=O

where in A)).

If X is an arbitrary subset of R”, a function X-t (wby definition is smooth if it is the restriction of a smooth function defined on some open set containing X’. The ring C”(X) of smooth functions on X is a Cm-ring. If X is closed, we find that every smooth function X-t II8 is the restriction of a smooth function defined on all of R” (smooth Tie&e), i.e., C”(R”) -+ C?(X) is a surjective C”-homomorphism. Consequently, if 1 is an ideal in CY’(!IP) and UC R” is an open set such that 3f~ I Z(f) c U then any ge C?‘(U) defines a unique element of the ring A = P(W),/1 (and analogously for A = Coo@“)/1 not necessarily finitely generated, and U the complement of a zeroset in IRE), We will often use this tacitly, or refer to this as smooth Tietze. Every C”-ring A has a canonical preorder < defined by sib

iff 3cg U(A),

c’=b-a.

If A = Cm( IRE)/1and f, g E C”( R”) then as elements of A, iff 39 E Z,Vx6 Z(cp), f(x) < g(x).

fO,

f+g>O,

with the ~ngstructure etc.

in the sense that

2. LOCAL Cm-RINGS

In this section we will discuss some genera1properties of local C”-rings. Our main purpose will be to give a direct proof of the fact that every local

RINGSOFSM~~F~CTIONS

329

P-ring is separably real closed (de~nitions will be given below). This result was first proved by different methods in Joyal and Reyes [S]. Any (Y-ring is in particular a commutative ring. A local C”-ring is a P-ring which is a local ring. (This definition differs from the often used, but confusing, terminology introduced in Dubuc [3]!) Important examples are rings of germs of smooth functions: if M is a manifold and p E i&f, the P-ring C; (M) of germs of smooth functions at p is local. Other examples can be obtained by taking quotients of rings of germs, such as formal power series and quotients of such (formal algebras). Indeed, the “Taylor series at W-map TO: c; fW -+ WEE,,..., x,11 is a surjective P-homomorphism by Borel’s theorem. Not every (finitely generated) local P-ring is quotient of a ring of germs. For instance, if P is a maximal (nonprincipal) filter on fV and Z= {f~ P’(fV)lZ(f)~g}, then Cm( N )/F is a local ring which is not a quotient of a ring of germs. 2.1. DEFINITION. A P-ring then A (a-’ 1 is nontrivial.

is called reduced if for every a E A, if a # 0

Of course, for Cm-rings it is not true that if A (a- I ) is trivial then a is nilpotent. If Z is an ideal in A, we define the C”-radical $ of Z by aE $

iff (A/Z) { a - ’ }

is trivial

iff 3bGZ,

b E U(A(a-‘}).

Z is called a P-radical ideal if Z= fi. For a P-ring A we write Aredfor A/ fi, which is a reduced C” -ring. So Zc A is P-radical iff A/I is reduced. Note that if 9: A + B is a homomorphism of P-rings and Jc B is a P-radical ideal, then cp-‘(J) is also P-radical. (For we have 9: A,@-‘(f)B/J, so if B/J is reduced then so is A/p-‘(J).) It will be useful to have a description of reduced P-rings in terms of “generators and relations.” If E is any set, and Zc P(W) is an ideal, there is a filter of zerosets in tw”,

(which is proper if Z is), and conversely, for any filter @ of zerosets in [WE there is an ideal @’= (fE cy nv)I Z(f) E CP)

330

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(which is proper if @ is). For an ideal Zc P(iR”), we call the ideal (1)’ the C”-radical of I, and again denote it by $. This is consistent with our earlier te~inolo~y, since 2.2. LEMMA. reduced ifs I=

Let A% Cm(RE)/Z be an arbitrary $1 in the sense that Z(g) = Z(f)

and

?-ring.

Then A is

fEI*gEI.

Proof: It s&ices to observe that for f~ C”(RE), A( l/f) is trivial iff IS SE (i)‘. But A(l/f) ’ t rivial iff 3g E I Z(g) c Z(f) (by the explicit description of A ( l/f) we gave in Sect. 1), iff ,fE (I)‘. 1

Note that from Lemma 2.2 it follows that a finitely presented P-ring, i.e., a ring of the form C~(~~)~(~,...,~~), is reduced iff it is point-determined, as defined, e.g., in Kock [7]. 2.3. LEMMA. Let A be a CcD-ring, and Ic A a prime ideal. Then $1 is also prime. Or in more algebraic terms, if A is a C”-domain, then so is A,,. Proof. We may restrict ourselves to the case where A = C”(RE). Clearly, if @ is a prime filter of zerosets in RE (i.e., a filter with Fw GE Q,+ FE Q, or GE @) then @ is a prime ideal. So we need to show that if Zc Cm(@) is a prime ideal, then iis a prime filter. Suppose F and G are zerosets in R” with Fu G = Z(q) and cpE I. We may assumethat rp2 0 (replace cp by 40~).Let D c E be a linite subset containing the coordinates involved in F, G, and q; i.e., there are closed sets F, i? c JR” and a smooth 3: lRD-+R such that q=@on,, F=n,‘(E), G=n;‘(G). Choose smooth nonnegative functions f, g: RD --) R such that Z(f) = F, Z(g) = z‘, and let Ifs=f--g: IFP-+ R. Consider the closed sets H= (x 1q?(x) d 01 and K= (xl e(x) 301. SincePu(7=2(@), HnZ(@)=Fand KnZ(4)=(?. So if we let h and k be nonnegative functions on RD with Z(h) = H and Z(k)=K, then Z(@+h)iF and Z(~$+k)=t?. But h.k=O, so (~,h)*(~+kk)07Eg=~‘+(P*(h+k).n,Ef. Iis prime, so ($++)07c,,E? or ((?,+ k) 0zD E 1, i.e., either FE 1 or GE l Thus f is a prime filter. 1 2.4. ~0~~~10~.

Every C”-domain

is local.

ProoJ It suffices to show this for finitely generated Cm-rings. So let A = Cm( R”)/Z be a domain. I is a prime ideal, hence the corresponding

filter f of closed sets is also prime (Lemma 2.3). Now let A g E Cm( IV) with invertible in A. Then 3Fo 1, VXE E’, S(x) +g(x) $0. Let f+s U=(xlf(x)#Of, Y= (x(g(x)#O). Then Fc: UuV, so by normality

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RINGS OF SMOOTH FUNCTIONS

there are closed G c U and H c V with F = G u H. i is prime, so either GE f or HE 1, i.e., either f or g is invertible in A. b The following proposition characterizes reduces P-domains. 2.5. ~OPOSITION. A C”-ring embeddable in a P-field.

is a reduced ~~-domain

iff it is C”-

Proof: (: is clear. For * we wish to construct a Cm-field in the usual way: let ,4, = A be the reduced P-domain under consideration, and let A n+ 1= A;, where for a reduced P-domain 3, B’ is the universal solution to inverting all nonzero elements in B. F= &, A, is a field. This proves + provided we can show that each A, --+A,,+, is an injective P-homomorphism. Arguing by induction, we prove that if B is a reduced P-domain, then so is B’ and B + B’ is injective. In fact this follows straightforwardly from the result of Muiioz and Ortega (Theorem 1.3). Indeed, write B’ = lixr~Bjd -I}, where the (filtered) colimit is taken over all finite subsets A c B- (Of, and B(d-‘) = B(b-‘) of course, if b is the product of the elements of A. Now suppose a E B, and a = 0 in B’. Then there is a b E B- (0) such that a=0 in B{b&‘}, so Bib-‘}(a-‘} =B{(a*b)-‘} is trivial. B is reduced, so a. b = 0 in B; and B is a domain, so a = 0 in B. Thus B -+ B’ is injective. To see that B’ is reduced, choose aE B’, a #O, and suppose B’fa-‘> is trivial. Then there is a bGB- (0) such that uE Bib-‘) and Bfb-‘)(a-“) is trivial. By Theorem 1.4 there are c, dE B with c invertible in B(b-‘> and a.c=d in B{b-‘}. So B{b&‘}{a-‘}zB{b*d-‘1, and hence d=O as before since B is a reduced domain, i.e., a = 0 in B(b - 1}. The proof that B is a domain is similar: we need to show that if a,,a,~B(b-‘)anda,~a,=Othena,=Oora,=OinB(b-’).Takeci,di with ci invertible in B(b-‘) and aj*ci=di in B(b-‘1. Then a,a,)-‘) is trivial, so bd,d,=O in B. B is a B{(bd,dJ’) = B(b-‘}(( domain, so dl=O or dZ=O. Thus a,=0 or a,=0 in B{b-‘}. B Our next aim is to show that local Cm-rings are Henselian. Recall that a local ring A with residue field k, is ~ense~~a~ if for every manic polymial p(t) with coefficients in A, simple roots of p in k, can be lifted to A, i.e., Va E k, (p(a) = 0 #p’(a) + 3a E A(p(a) = 0 in A & n(a) = c(j),

where II denotes the canonical map A -+ k,. (Such a lifting is necessarily unique.) For general information see Raynaud [13 J We will need the following version of the implicit function theorem. 2.6. LEMMA. (IFT for closed sets). Let f (x, t): ET” x II4-+ R be smooth, and let Fc 08” be closed. Suppose y: IF!“’-+ R is a smooth function such that

332

MOERDIJK AND REYES

f(x, y(x)) = 0 # (af/at)(x, y(x)) for all XE F. Then there exists an open UXF and a tube

B= ((x, t)E Ux RI It-y(x)1 0 such that for each x E U,,, f (x, - ) has exactly one zero in (y(xO) - 6,, y(xO) + 6,,). Let V= lJxgErUrO. By a partition of unity argument, we can find a smooth p: V-+ (0, co) c [wsuch that if XE V then f (x, - ) has at most one zero in (y(x) -p(x), y(x) + p(x)) (making V a little smaller if necessary). Let U = {x E Vlf (x, - ) has exactly one zero in (y(x)-p(x), y(x) + p(x))}. Then UzF, and by the implicit function theorem, U is open and the function on U which associates with x E U this unique zero is smooth. 1 2.7. THEOREM. Let A and B be Co3-rings, with B reduced, and let cp:A-++B be a surjective C”-homomorphism which is local (i.e., cp(a)E U(B)+ae U(A)), and let p(t) be a manic polynomial in Art]. Then any simple root of p(t) in B can be ltyted to a root in A.

Proof: Choosing a set E of generators for A, we can write A = P(R”)/1 and B = Cuo(RE)/J, where J11 and cp is the canonical quotient map. Let @= { Z(f )If E I} be the filter of zerosets corresponding to J, so f EJ iff Z(f) E @ since B is reduced (Lemma 2.2). Suppose p(t) is represented by f(x,

with fi (x) E P(R”),

t)=t”+f,(x)

t”-‘+

and let r E P(R”) f (x, r(x)) = 0

$4

r(x)) Z 0

... +f,(x),

be a simple root in B, that is, on some G E @, on some HE @.

Let F = G n H, and let D c E be a finite set containing all the coordinates involved in the fi(x), r(x), and F. So we can regard f (x, t) as a function RD x R -+ R, r(x) as RD + R, and F as a closed subset of RD, and we have f(x, r(x))=OZ$x,

r(x)),

QXEF.

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RINGSOF SMOOTH FUNCTIONS

By Lemma 2.6, there is an open UIF and a smooth s: U -+ 58 with s 1F= r 1F (so q(s) = r since B is reduced), and f(x, s(x)) = 0 for all x E U. We wish to conclude that p(s) = 0 in A, i.e., thatf(x), f(x)) E I. Let g(x) be a characteristic function for U. Then g(x) .f(x, s(x)) = 0 in Cm(!P), hence in A. But g is invertible in B since Fc U and B is reduced, so g is invertible in A, and therefore f(x, s(x)) = 0 in A. 1 Applying Theorem 2.7 to the special case where A is a local C”-ring and l3 is its residue field, we obtain 2.8. COROILARY.

Every local C”-ring

is Henselian.

4

It should be observed that in the proof of 2.7 we did not use that the function f(x, t) representing p(x) depended polynomially on t. So the argument remains valid if we assume p E A( t > rather than p E Art]. Rewriting the definition of Hensehan local ring with A( t > instead of Act] gives a notion which is more natural in the context of P-rings, and which we call C”-Henselian. Thus, as a strengthening of 2.8 we have 2.8’. COROLLARY.

Every local C”-ring

is C~-Henselian.

[

2.9. COROLLARY. For every local Y-ring A we have (as rings) that AZ kA Qm,, where k, is the residue field and mA is the maximal ideal. Proof: We show that the exact sequence 0 -+ mA + A + kA -+ 0 is splitexact. Consider partial sections (K, f) of 7~,where K is a subfield (Ralgebra) of k,. Let (K, s) be a maximal section (Zorn); A

Take a E kA - 1%If a is transcendental over K, we can extend s to a section on K(M) = K(x), contradicting maximality. And if LXis algebraic over K, there is an irreducible manic polynomial f with f(ol) = 0, f'(a) # 0. By Henselianness,a can be lifted to a root /3E A, x(B) = a, an by sending c1to fl we obtain an extension of s to K(a), again contradicting maximality of s. So K=k,.

1

Recall from section 1 that every (Y-ring has a canonical pre-order < . If A = Cm([WE)/I,then for f E Coo(lR”) representing an element of A, 0-O

in A iff 3g E 1, ‘dx E Z(g), j”(x) > 0.

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MOERDIJK AND REYES

If A is a field, I is maximal (hence I= fi), and the pre-order is a total order, i.e., f#O-+fO. A totally ordered field is called real closed if it satisfies (a) x>O=+3yx=y2, (b) polynomials of odd degree have roots. 2.10. THEOREM. Every Cm-field is real closed. Proof: Let K= P(UX”)/Z be a P-field, and let @ be the maximal filter of zerosets corresponding to the maximal ideal I. We have just remarked that K is totally ordered in a canonical way. Now condition (a) is trivial, and holds in fact in any P-ring (use smooth Tietze, Sect. 1). To prove condition (b), let p(t) E K[t] be a polynomial of odd degree. We may assume that p is manic and irreducible, and hence that (p, p’) = (1) as ideals in K[t]. Or equivalently, the resultant determinant Res(p, p’) # 0 in K. Therefore, if p(t) is represented by f(x, t)= t”+f,(x)

t”-‘+

... +f,(x),

then for the function R(x) = Res(f (x, t), af/at(x, t)), we have that R(x) # 0 for all x in some FE @. As before, choose a finite D c E containing all the coordinates involved in f and in F, and regard f as a function [wDx Iw+ Iwand F as a closed subset of RD. Let U c IwDbe open, Fc U, such that R(x) # 0 on U. R! is real closed, so for each x E U the polynomial f (x, t) E R[t] has a root. Let r(x) be the first root. Then this root is simple since R(x) # 0, so r: U + R is smooth by the implicit function theorem (the usual, not 2.6). r represents an element of K which is a root of p. 1 A local ring A is called separably real closed if A is Henselian and k, is real closed. So combining 2.8 and 2.10 we have 2.11.COROLLARY.

Every local ?-ring

is separably real closed.

1

Although 2.11 has been proved for P-rings in the topos of Sets, if follows that 2.11 is true for local P-rings in any (Grothendieck) topos, since the notions involved are all coherent, (see Kock [6] and Joyal & Reyes [S]), so we can apply the completeness theorem of Makkai and Reyes [S]. As with Henselianness, we would like to define a “smooth” notion of real-closedness, using K(t) instead of K[t], which implies the usuul algebraic notion. Analysing the proof of 2.10 we see that we need is to

335

RINGS OF SMOOTH FIXKTIONS

replace condition (b) that odd polynomials have roots by a “transversal intermediate value” condition: trf~K{t) ((f(O)*f(l)